Symplectic Twist Maps

Zero entropy and stable rotation sets for monotone recurrence relations

“…It is easy to check that B ⊂ B, B is closed in the product topology, and τ m,n B = B, for all m, n ∈ Z. It follows that each Birkhoff configuration has a rotation number [6,15].…”

“…There are several objects called Birkhoff in this section. The notion of Birkhoff configuration, which is central in this paper, corresponds to that of Birkhoff orbit, which plays an important role in the Aubry-Mather theory, see, for example, [3,6,15]. Another terminology, Birkhoff recurrence class, was borrowed from [24].…”

“…Let be a Birkhoff configuration. Then has a rotation number and Moreover, the map is continuous in the product topology [6, 15].…”

Monotone gradient dynamics and the location of stationary (p,q)-configurations

“…Angenent [1] was the first to exploit the monotonicity of this gradient semiflow in the study of periodic orbits of twist maps. Golé [10], Mramor and Rinky [15] and Slijepčević [17] have also used the monotonicity of the gradient semiflow of the FK-model to prove the existence of ghost circles and to prove the existence of Mather's shadowing orbits of twist maps, respectively.…”

“…We consider an exact C r -area preserving positive twist diffeomorphism (r 1), f , of the infinite annulus A = T 1 × R, as represented by a lift, F (for basic properties of such maps the reader is referred to [10]). π : R 2 → A denotes the covering projection.…”